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Free Vibrations of Circular Cylindrical Shells

Tatsuzo Koga

e-mail: [email protected]

URL: http://www.geocities.co.jp/SiliconValley-Bay/1245

Abstract

The eigenvalue problem of the free vibrations of thin elastic c ircularcylindrical shells is one of the well-established classical topics in structural

mechanics. All the characteristic values of interest can be calculated

nowadays to a desired degree of accuracy as a routine work with the

aid of digital computers. A number of analytical solutions have been

proposed, which may help to gain a good insight into the physical nature

underlying the flood of numerical data poured from the computers. This

paper reviews the historical background and provides a unified view of

the current state of the art through asymptotic solutions recentlyobtained by the author. Emphasis is placed on the effects of the

boundary conditions.

1. Inextensional Vibrations

It was more than a hundred years ago that a valid solution for the

free vibrations of thin elastic circular cylindrical shells was first given by

Lord Rayleigh, with the birth name J ohn William Strutt. In his famoustreatise on "The Theory of Sound"(1) published in 1877, Lord Rayleigh

derived an expression for the natural frequency of an infinitely long

cylindrical shell, assuming that the midsurface of the shell admits no

stretching and that the mode shape is cylindrical. Lord Rayleigh's solution

for the natural frequency, f R, is given by

f R2 = Eh2n2(n2-1)2/48ƒ Î 2(1-ƒ Ë 2)ƒ Ï R4(n2+1) (1)



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where h is the thickness, R the radius, E Young's modulus, ƒ Ë Poisson's ratio,

ƒ Ï mass per unit volume, and n the circumferential wave number. The

mode shape is represented by the lateral deflec tion w such that

w = W0 cos nƒ Æ sin 2ƒ Î f Rt (2)

where W0 is an arbitrary constant, ƒ Æ the circumferential coordinate, and

t the time. Rayleigh's solution also holds for the inplane vibrations of a

circular ring. Indeed, he showed that it is identical in form with the one

derived by Hoppe for a ring in a memoir published in Crelle, Bd.63, 1871.

In 1881, Lord Rayleigh presented a paper(2) in which he developed a

theory of the inextensional vibrations for shells of revolution and applied it

to shells of spherical, conical and cylindrical shape. His primary interest in

this paper seems to have been in the calculation of natural frequencies

of hemispherical shell as a model of a church bell, because a substantial

part of the paper consists of the analysis of this problem.

Love presented a paper(3) in 1888 (received J anuary 19,read February

9), in which he criticized Lord Rayleigh's theory as inexact in that it leads

to an expression for the displacements which can not satisfy the

boundary conditions at the free ends. He argued that the equationsimposing the condition of inextensibility are, in the most general case, a

system of the third order, while the boundary conditions are four in

number, and hence these equations are not, in general, of a sufficiently

high order to admit of solutions which satisfy the boundary conditions at

the free ends. He explicitly showed that this is the case for a spherical shell.

He has also derived a strain energy expression which consists of two parts;

one due to membrane stretching and proportional to the thickness h and

the other due to flexural bending proportional to h3. He argues that theterm proportional to h3 is small for thin shells in comparison with the term

proportional to h, and that the former instead of the latter should be

omitted in the limit of h•̈ 0. He has thus formulated a theory for the

extensional vibrations and applied it to spherical and cylindrical shells.

Lord Rayleigh responded immediately to Love's criticism in a paper(4)

presented in the same year (received December 1, read December 19).

He states that it is a general mechanical principle that, if displacements

be produced in a system by forces, the resulting deformation is



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determined by the condition that the potential energy of deformation

shall be as small as possible, and that the large potential energy which

would accompany any stretching will not occur. He gives an explanationto the violation of the boundary conditions at free ends by means of an

example of a long cylinder subjected to a pair of concentrated normal

forces at the extremities of a diameter of the cross sec tion at the center

of the shell. As the thickness reduces, the deformation assumes more and

more the character of pure bending such that every normal cross section

deforms into an identical configuration. If the thickness remains small but

finite, a point will at last be attained when the energy can be made least

by a sensible local stretching of the midsurface such as will dispense with

the uniform bending otherwise necessary over so great a length.

The second edition of "The Theory of Sound" was published in 1894. All

the corrections of importance and new matter added to the first edition

are c learly indicated. A new chapter is incorporated devoted to shells, in

which it is noted that any extension that may occur in the inextensional

vibrations must be limited to a region of infinitely small area adjacent to

the ends and affects neither the mode nor the frequency of the

vibrations. The first edition of Love's treatise on "The Mathematical Theory

of Elasticity"(5) was published in 1892 (volume 1) and 1893 (volume 2). It'sfourth edition was published in 1927, whose American printing in 1959 is

the one available to the author at the present time. Love seems to have

acknowledged Rayleigh's theory, and he notes that the membrane strain

which is necessary in order to secure the satisfaction of the boundary

conditions is practically confined to so narrow a region near the ends that

its effect in altering the total amount of the potential energy is negligible

and a greater part of the shell vibrates without accompanying it. Love

himself has derived a solution of the inextensional vibrations of acylindrical shell of finite length having free ends. Love's solution differs

from Rayleigh's in that the vibration mode is linear along the generator

and asymmetric about the cross section bisecting the axis and also in

that it contains a length parameter. The natural frequency of Love's

solution, f L, is given by

f L2 = f R2{1+6(1-ƒ Ë )R2/n2L2}/{1+3R2/n2(n2+1)L2} (3)



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where L is one half the length of the cylinder. It should be noted that f L is

always higher than f R but the difference is very small when the shell length

is greater than or about the same as the radius. The mode shape isrepresented by a linear function of the axial coordinate x.

w = W0 x cos nƒ Æ sin 2ƒ Î f Lt (4)

The issue seems to have been settled. State of the art remains essentially

the same for a century. It has been widely accepted that the

inextensional vibrations occur either in Rayleigh's or Love's mode either

when the cylinder is indefinitely long or when both ends are free for a

finite cylinder. Many have reported that they have detected the

inextensional vibrations and measured the resonance frequencies, but

none of them seems to have identified the resonance modes.

In 1980, Koga(6) suggested a possibility of yet another mode of the

inextensional vibrations when one end of the cylinder is free and the

other is supported in such a manner that it can move freely in the axial

direc tion. Subsequently, Koga and his collaborators(7,8) have shown by

asymptotic method that the inextensional vibrations can occur in three

distinct modes; Rayleigh's, Love's and a new mode, at a frequencyapproximately equal to f R, and that the mode shape function of the new

mode is represented by a linear combination of those of Rayleigh's and

Love's, such that

W = W0 (x+L) cos nƒ Æ sin 2ƒ Î f Rt (5)

where x= L corresponds to the free end and x= - L to the supported end.

Typical mode shapes of these inextensional vibrations are depictedschematically in Fig.1.

Koga and his collaborators conducted an experiment and have

proved the existence of these three modes by detecting and identifying

their modes in resonance. In the experiment, the mode shapes were

visualized by holographic interferometry. Typical fringe patterns of the

resonance modes are shown in Fig.2. Both Rayleigh's and Love's modes

were observed in a test specimen having completely free ends. The



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specimen was attached by a supporting column by a small screw at a

point bisecting the axis of the cylinder. The new mode was observed in a

test specimen which had a free end and a supported end. Thesupported end was fitted with a diaphragm made from a flexible thin

annular plate whose inner circle was fastened to a supporting column by

a pair of nuts. The cross sections of the test specimens are shown in Fig.3.

Fig.1. Free Vibration Modes: (a)Rayleigh mode, (b)Love mode,

(c)new mode



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Fig.2. Fringe Patterns of Rayleigh, Love, and new modes



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Fig.3. Cross Sections of Test Specimens: (a)FR-FR, (b)FR-SF

2. General Flexural Vibrations

Love suggested in Ref.5 a method for solving eigenvalue problems of free

vibrations in general modes which may accompany a slight midsurface

stretching, but he had made no attempt to apply it to any specific

problem. Flugge published a book(9) in 1934, in which he has established

a general method of solution for the free vibrations of circular cylindrical

shells. Flugge's method may be summarized as follows:

Let the axial, circumferential and lateral displacements be denoted by u,

v and w, and let them be assumed in the form

u = U0 exp (px) cos nƒ Æ sin 2ƒ Î ft

v = V0 exp (px) cos nƒ Æ sin 2ƒ Î ft (6)

w = W0 exp (px) cos nƒ Æ sin 2ƒ Î ft

where U0, V0 and W0 are arbitrary constants. Substitution of Eqs.(6) into the

equations of motion results in the auxiliary equation which is an algebraic

equation of the fourth degree in p2 and n2 and of the third degree in f 2.



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Thus, if values of n and f are assumed, eight roots for p may be

determined. Once the values of p, n and f as well as the

thickness-to-radius ratio and the material constants are known, thecharacteristic equations resulting from consideration of the boundary

conditions may be solved for the length of the cylinder 2L. The minimum

of them determines the length of the cylinder which vibrates at f in a

mode having 2n of nodal lines along the generator and none around the

circumference.

Calculation of numerical solutions by Flugge's method is tedious in

general without the use of high speed digital computers. Flugge could

only applied it to a special case where the ends are simply supportedsuch that

w = Mx = Nx = v = 0 ; at x = L and x= -L (7)

where Nx is the axial stress resultant and Mx is the axial stress couple. In this

special case, the exponential functions in Eqs.(6) can be represented by

trigonometric functions such that

u = U0 sin ƒ Î (x+L)/2L cos nƒ Æ sin 2ƒ Î ft

v = V0 cos ƒ Î (x+L)/2L sin nƒ Æ sin 2ƒ Î ft (8)

w = W0 sin ƒ Î (x+L)/2L cos nƒ Æ sin 2ƒ Î ft

It can be shown easily that Eqs.(8) satisfy Eqs.(7) exactly. The auxiliary

equation can now be solved for f for given values of n.

Flugge calculated three roots of f for a specific numerica l example.

Comparing the amplitude ratios of the displacement components for

each root, he has shown that the free vibrations corresponding to the

lowest, intermediate and highest values of f are predominantly in lateral,

axial and circumferential motion, respectively. He has derived a simple

formula for the lowest natural frequenc ies neglecting in the auxiliary

equation cubic and quadratic terms in f 2.



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Much of the research effort thereafter and before the emergence

of digital computers was direc ted toward obtaining approximate

solutions and verifying them by experiment. Boundary conditionsconsidered in those early days are those of the simply supported ends

defined by Eqs.(7), the rigidly clamped ends defined by

w = w' = u = v = 0 ; at x = L and x = -L (9)

or the free ends defined by

Qx = Mx = Nx = Tx ƒ Æ = 0 ; at x = L and x = -L (10)

where w' is the partial derivative of w with respec t to x, and Qx and Tx ƒ Æ

are the lateral and circumferential components of the equivalent

edge-shear. Arnold and Warburton(10) calculated the natural frequencies

of circular cylindrical shells of finite length, which are either simply

supported or rigidly clamped at both ends. An equivalent wave length

has been introduced in order to dea l with cylinders having solidly built-in

or flanged ends. In experiment, the modal characteristics were

determined by detecting the sound intensity variations by stethoscope.Baron and Bleich(11) calculated the natural frequencies of an infinitely

long cylindrical shells by Rayleigh's method. Yu(12) used Donnell type of

approximate equations to calculate the natural frequencies of cylindrical

shells which were either simply supported or rigidly clamped at both ends,

or simply supported at one end and rigidly clamped at the other.

Assuming that the shell is relatively long and vibrates in a mode

characterized by a small number of the longitudinal waves and a large

number of the c ircumferential waves, he has derived a simple formula forthe natural frequency. The characteristic equations turned out to be

identical with those of an elastic beam in lateral vibrations. Yu's method

was used by Smith and Haft(13) and Vronay and Smith(14) to calculate the

natural freqenc ies of cylindrical shells having rigidly clamped ends. The

Donnell type of approximation was also used by Heki(15). From an analysis

on the characteristic values of both the circumferentially closed and

open shells, Heki has drawn the following conc lusions: (i) The Donnell type

of approximation gives accurate solutions for natural frequency provided



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that the frequency is not very high and the shell is not extremely long. (ii)

The edge-zone bending solutions which decay out rapidly as the

distance from the ends increases do not have significant influence on thenatural frequency. (iii) The natural frequency is governed predominantly

by the global solutions which vary gradually over the entire surface of the

shell and affected significantly by the inplane boundary conditions.

Weingarten(16) made an analysis in the case where one end is rigidly

clamped and the other is free and conducted an experiment on

butt-welded steel test specimens using a microphone. Watkins and

Clary(17) conducted an experiment on spotwelded stainless steel test

specimens having the free-free and the c lamped-free ends. Sewall and

Naumann(18) conducted an analytical and experimental study on the

free vibrations of unstiffened shells. The unstiffened test specimens were

fabricated from aluminum alloy sheet. Four sets of boundary conditions

were considered; the free-free, the simply supported-simply supported,

the c lamped-free, and the c lamped-clamped ends. Nau and

Simmonds(19) have derived a simple formula for the low natural frequency

of a clamped-clamped shell by asymptotic method.

In the meantime, the development and rapid progress of digital

computers enabled researchers to calculate more accurate solutionsnumerically. The accuracy and the range of validity of the approximate

solutions have been examined by comparing with more accurate

solutions of the classical shell theory or with elasticity solutions.

Greenspon(20) calculated the natural frequency of the flexural vibrations

of a hollow cylinder of finite length considering it as a three dimensional

body. The results were compared with those of the Timoshenko beam

theory in the case of n=1 and with those calculated by Arnold and

Warburton(10) in the case of n=2. Gazis(21) made an analysis by elasticitytheory on the plane-strain free vibrations of a thick-walled cylinder of

finite length. He has shown that the extensional and shear modes can

exist uncoupled in the axisymmetric vibrations and derived approximate

expressions of the frequencies for these modes, which approach those of

the simple thickness-stretch and thickness-shear modes of an infinite plate

as the thickness-to radius ratio approaches zero. The minimum of them is

given by f s of the simple thickness-shear mode:



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f s2 = E/8(1+ƒ Ë )ƒ Ï h2 (11)

Armenakas(22)

calculated the natural frequencies of a simply supportedshell by the Flugge and the Donnell type equations and has established a

range of validity of these equations on the basis of numerical comparison

with the results of Refs.20 and 21. Sharma and J ohns(23) calculated the

natural frequencies of a finite shell whose ends are either rigidly clamped

and free or rigidly clamped and ring-stiffened. The results show that there

is no significant difference between the natural frequencies calculated

by Flugge theory and those by Timoshenko's version of the Love theory.

It is well anticipated, as noted by Kalnins(24), from the results of Gazis'

analysis that the solutions for the natural frequency f calculated by the

classical theory are valid in a spectrum well below f s. Since the classical

theory is established on the basis of the fundamental assumption

h/R << 1 (12)

it may be stated that it is valid when

f/f s = O(h/R) (13)

Furthermore, Eqs.(1) and (2) imply that the natural frequency of highly

flexural modes is as low as

f/f s = O[(nh/R)2] (14)

provided that nh/R is much smaller than unity. The last condition always

holds for the classical theory of thin shells: Niordson(25) has proved that thefundamental assumptions underlying the c lassical theory are equivalent

to stating that

(h/ƒ É m)2 << 1 (15)

where ƒ É m is the minimum wave length of deformation or vibrations.



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3. Effects of Boundary Conditions

Arnold and Warburton(10) made a systematic attempt to clarifying the

effec ts of the boundary conditions by calculating approximate solutions

under various boundary conditions of practical interest and by

conducting an experiment. They were followed by many researchers,

Refs.11-19, who calculated the natural frequencies under the boundary

conditions prescribed by Eqs.(7), (9) or (10). As a result, it became c lear

that the natural frequency depends on these three sets of boundary

conditions. The importance of the inplane boundary conditions was firstnoted by Heki(15) in his memorable but unduly overlooked paper. The

significance of the effect of the inplane boundary conditions became

more convincing by an extensive numerical analysis of Forsberg(26,27).

In principle, in the framework of the classical theory of thin shells, the

boundary conditions at an end of a circular cylindrical shell can be

imposed by an appropriate combination of the following four pairs:

w = 0 or Qx = 0w' = 0 or Mx = 0

u = 0 or Nx = 0 (16)

v = 0 or Tx ƒ Æ = 0

There are sixteen possible combinations. Forsberg solved Flugge's

equations numerically in all sixteen cases of the boundary conditions in a

broad range of geometric parameters. He has presented the results only

in ten representative cases including those specified with different

boundary conditions at opposite ends. One of the most important

conclusions of his analysis on the breathing vibrations ( n>2) is that the

effect of the axial constraint is significant even for very long shells for all

values of the thickness-to-radius ratio, so that the minimum frequencies

differ by more than 50 per cent depending on whether u=0 or Nx=0 is

imposed at both ends. As to the axisymmetric ( n=0) and the beam-like

bending (n=1) vibrations, he has shown that the vibration characteristics

are strongly influenced by the c ircumferential as well as the axial

constraints.



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The simply supported and the rigidly clamped ends are often used in

engineering practice as idealized mathematical models for the

supported ends of real shell structures which fall in somewhere betweenthe two models. There are four variations for each of these models

depending on the inplane boundary conditions. They are designated

and defined as follows:

For simply supported ends

S1: w = Mx = u = v = 0

S2: w = Mx = u = Tx ƒ Æ = 0

S3: w = Mx = Nx = v = 0 (17)

S4: w = Mx = Nx = Tx ƒ Æ = 0

For clamped ends

C1: w = w' = u = v = 0

C2: u = w' = u = Tx ƒ Æ = 0

C3: u = w' = Nx = v = 0 (18)

C4: u = w' = Nx = Tx ƒ Æ = 0

Yamaki(28) calculated numerical solutions for the eigenvalue problemsunder various combinations of the boundary conditions of Eqs.(17) and

(18). His results confirm Forsberg's conclusions on the effec ts of the axial

constraints.

More recently, Koga (29) have derived asymptotic solutions dealing with

all possible combinations of the boundary conditions for the free ends,

Eqs.(10), the simply supported ends, Eqs.(17), and the clamped ends,

Eqs.(18). If the set of the boundary conditions defined by Eqs.(10) is

designated by FR, the asymptotic solutions are valid for all forty-fivepossible combinations between the opposite ends of the nine sets

consisting of S1, S2, S3, S4, C1, C2, C3, C4, and FR. The asymptotic solution

for the natural frequency is given by

f 2 = f R2[ 1 + 12(1-ƒ Ë 2)R2 ƒ Ì 4/h2(n2-1)2 ] (19)

where f R is Rayleigh's solution given by Eq.(1), and ƒ Ì is the eigenvalue to

be determined from consideration of boundary conditions. Equation (19)



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is identical in form with that derived by Nau and Simmonds(19) for the

cases of C1-C1. The characteristic equations to be satisfied by ƒ Ì have

been obtained for all forty-five combinations of the boundary conditions.It turned out that it suffices for deriving the characteristic equations to

specify only three sets of representative boundary conditions defined

and desiganated as

SR: w = u = 0

SF: w = Nx= 0 (20)

FR: Nx = Tx ƒ Æ = 0

Table 1. Characteristic Equations and Combinations of

Boundary Conditions

There are six combinations of these sets between two ends; SR-SR, SR-SF,

SF-SF, SR-FR, SF-FR, and FR-FR. The last two result in the solutions for the

inextensional vibrations described in Section 1. The remaining four sets



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result in four mutually independent characteristic equations which are

identical in form with those for the free vibrations of an elastic beam. The

characteristic equations and the combinations of the boundaryconditions are summarized in Table 1.

The representative boundary conditions, Eqs.(20), indicate clearly that

the free vibration characteristics of a circular cylindrical shell depend on

whether the ends are free or supported, and whether the supported ends

are axially constrained.

The minimum roots of the characteristic equations are given by

2nLƒ Ì /R = 4.730 ( Type I ),

= 3.927 ( Type II ),

= 3.142 ( Type III ),

= 1.875 ( Type IV ),

= 0 ( Type V ) (21)

For given values of L/R and n, the values of ƒ Ì are determined from

Eqs.(21), which are then substituted in Eq.(19) to calculate the natural

frequencies. Results are shown accurate enough for engineering

purpose on the basis of a comparison with more accurate numerical

solutions. This has been also proved by an experiment. In the test

spec imens, the representative boundary conditions of SR were

established by soldering an end to a metal block by a low melting point

metal, those of SF by attaching a flexural annular plate at an end byadhesive resin, and those of FR by leaving an end completely free. The

cross sections of the test spec imens of Type I, II, III, and IV are depicted

in Fig.4. Test results show a satisfactory agreement with the theoretical

preditions by Eq.(19).



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Fig.4. Cross Sections of Test Specimens of Type I, II, III, IV

4. Some Related Eigenvalue Problems

In the theory of thin-walled structures, surface tractions acting on the wall

surfaces and body forces in the volume of mass points are treated alike

by replacing them by the equivalent forces acting on the reference

surfaces. It can be anticipated very well therefore that the analyses and

conclusions of the preceding sections for the free vibrations can readily

be extended to include the effect of pressurization. If the inertial force



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vanishes and only the external pressure is present, the analyses will

provide solutions for the bifurcation buckling. When the cylindrical shell is

filled with a liquid, the fructuating pressure exerted upon the shell from theoscillating liquid may be incorporated in the shell body as inertial force

acting on virtual mass. Then, again, the asymptotic method and solutions

of the preceding sections can readily be applied to this problem to

obtain the natural frequency of a liquid-filled shell.

Koga and his collaborators(30,31,32,33) have obtained solutions for these

problems.

Natural frequency under pressure p:

f 2 = f R2[ 1 + 12(1-ƒ Ë 2)R2 ƒ Ì 4/h2(n2-1)2] + n2(n2-1)p/4ƒ Î 2 ƒ Ï hR(n2+1) (22)

where p is positive for internal pressure.

Buckling pressure:

p = - [ Eh3(n2-1)/12(1-ƒ Ë 2)R3 ]{1 + 12(1-ƒ Ë 2)R2 ƒ Ì 4/h2(n2-1)2 } (23)

Natural frequency of a liquid-filled shell:

f 2 = {Eh2n2(n2-1)2/48ƒ Î 2(1-ƒ Ë 2)R4[(ƒ Ï +ƒ Ï v)n2 + ƒ Ï ] }{1 + 12(1-ƒ Ë 2)R2 ƒ Ì4/h2(n2-1)2 } (24)

in which rv is the virtual mass per unit volume and is given in terms of the

liquid-mass per unit volume, rL, approximately by

ƒ Ï v = ƒ Ï LR/nh (25)

References

(1)Lord Rayleigh,The Theory of Sound,2nd. Ed. Dover Publicati ons,1945

(1st. Ed.1877,2nd. Ed. 1894)

(2)Lord Rayleigh,On the Infinitesimal Bending of Surfaces of R evolution,

Proc. of London Math. Soc.,Vol.13,Nov. 1881,pp.4-17

(3)Love,A.E.H.,The Small Free Vibrations of and Deformation o f a Thin

Elastic Shell,Phil. Trans. Roy. Soc. of London,Ser.A,Vol.179, Feb.1888,

p.491-546



18

(4)LordRayleigh,OntheBendingandVibrationofThinElastic Shells,

EspeciallyofCylindricalForm,Proc. OfRoy.Soc.OfLondon,Vo l.45,Dec.

1888,pp.105-123(5)Love,A.E.H.,ATreatiseontheMathematicalTheoryofElast icity,"

4th.Ed.,CambridgeUniv.Press,AmericanprintingbyMacmillan Co.,1959

(1st.ed.1892 and1893,2nd.Ed.1906,3rd.Ed.1920,4th.Ed. 1927)

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July1986,pp.166-169 (inJapanese)

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1962

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Soc.ofLondon,SeriesA,Vol.197,No.1049,June1949,pp.283-25 6

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Mech.,Vol.21,No.2,June.1954,pp.178-184

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(13)Smith,B.L.andHaf t,E.E.,NaturalFrequenciesofClamped Cylindrical

Shells,AIAAJournal,Vol.6,No.4,April1968,pp.720-721

(14)Vronay,D.F.andSmith,B.L.,FreeVibrationofCircularCy lindricalShellsofFiniteLength,AIAAJournal,Vol.8,No.3,March1970, pp.601-603

(15)Heki,K .,VibrationofCylindricalShells,JournalofInst. of

Polytechnics,OsakaCityUniv.,SeriesF,Vol.1,No.1,November 1957

(16)Weingarten,V.I.,FreeVibrationofThinCylindricalShells ,AIAA

Journal,Vol.2,No.4,April1964,pp.717-722

(17)Watkins,J.D.and Clary,R.R.,VibrationalCharacteristics ofSome

Thin-WalledCylindricalandConicalFrustumShells,NASATND-27 29,March

1965



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(18)Sewall,J.L.andNaumann,E.C.,AnExperimentalandAnalyti cal

VibrationStudyofThinCylindricalShellsWithandWithoutLong itudinal

Stif f eners,NASATND-4705,September1968(19)Nau,R.W.andSimmonds,J.G.,CalculationoftheLowNatura l

FrequenciesofClampedCylindricalShellsbyAsymptoticMethods, Int.

JournalofSolidandStructures,Vol.9,No.3,May1973,pp.591-6 05

(20)Greenspon,J.E.,FlexuralVibrationsofaThinWalledCircu lar

Cylinder,Proc.of3rd.U.S.NationalCongressofAppl.Mech.,J une1958,

pp.163-173

(21)Gazis,D.C.,ExactAnalysisofPlane-StrainVibrationsofT hick-Walled

HollowCylinders,JournalofAcoust.Soc.ofAmerica,Vol.30,No .8,August

1958,pp.786-794

(22)Armenakas,A.E.,OntheAccuracyofSomeDynamicShellTheo ries,

JournalofEng.Mech.Div.,Proc.ofAmericanSoc.ofCivilEng. ,Vol.93,

EM-5,October1967,pp.95-109

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